Electronics Handbook/Circuits/Parallel Circuit

Series Circuit
Electronic components R,L,C can be connected in parallel to form RL, RC, LC, RLC series circuit
 * 1) /RC Parallel/
 * 2) /RL Parallel/
 * 3) /LC Parallel/
 * 4) /RLC Parallel/

Parallel RC

 * [[Image:RC switch.svg|200px]]

The total Impedance of the circuit
 * $$Z = Z_R + Z_C = R + \frac{1}{j\omega C} = \frac{1 + j\omega RC}{j\omega C}$$
 * $$Z = \frac{1}{j\omega C} (1 + j\omega T$$)
 * T = RC

At Equilibrium sum of all voltages equal zero
 * $$C \frac{dV}{dt} + \frac{V}{R} = 0$$
 * $$\frac{dV}{dt} = - \frac{1}{RC} V$$
 * $$\frac{1}{V} dV = - \frac{1}{RC} dt$$
 * $$\int \frac{1}{V} dV = - \int \frac{1}{RC} dt$$
 * ln V = $$ - \frac{1}{RC} + C$$
 * $$ V = e^- (\frac{1}{RC}) t + C$$
 * $$ V = A e^- (\frac{1}{T}) t $$
 * $$A = e^C$$
 * T = RC

Circuit's Impedance in Polar coordinate
 * $$Z = Z_R + Z_C $$
 * $$Z = R \angle 0 + \frac{1}{\omega C} \angle - 90 $$
 * $$\sqrt{R^2 + (\frac{1}{\omega C})^2} \angle Tan^-1 \frac{1}{\omega RC}$$

Phase Angle Difference Between Voltage and Current ''There is a difference in angle Between Voltage and Current. Current leads Voltage by an angle θ''
 * $$Tan\theta = \frac{1}{\omega RC} = \frac{1}{2\pi f RC} = \frac{1}{2\pi} \frac{t}{T}$$

Summary
RL series circuit has a first order differential equation of voltage
 * $$\frac{d}{dt}f(t) + \frac{t}{T} = 0$$

Which has one real root
 * $$V(t) = Ae^\frac{-t}{T}$$
 * $$A = e^c$$

The Natural Response of the circuit at equilibrium is a Exponential Decrease function

Phase Angle Difference Between Voltage and Current
 * $$Tan\theta = \frac{1}{\omega RC} = \frac{1}{2\pi f RC} = \frac{1}{2\pi} \frac{t}{T}$$

Parallel RL


The total Circuit's Impedance In Rectangular Coordinate
 * $$Z = Z_R + Z_L = R + j \omega L $$
 * $$Z = \frac{1}{R} (1 + j\omega T) $$
 * $$T = \frac{L}{R}$$

At Equilibrium sum of all voltages equal zero
 * $$L\frac{dI}{dt} + IR = 0$$
 * $$\frac{dI}{dt} = - I \frac{R}{L}$$
 * $$\int \frac{1}{I} dI = - \int \frac{L}{R} dt$$
 * ln I = $$(-\frac{L}{R} + c)$$
 * I = $$e^(-\frac{L}{R} + c) t$$
 * I = $$e^c e^(-\frac{L}{R}t)$$
 * I = $$A e^(-\frac{L}{R}t)$$

Circuit's Impedance In Polar Coordinate
 * $$Z = Z_R + Z_L = R \angle 0 + \omega L \angle 90$$
 * $$\sqrt{R^2 + (\omega L)^2} \angle Tan^-1 \omega\frac{L}{R}$$

Phase Angle of Difference Between Voltage and Current
 * $$Tan \theta = \omega \frac{L}{R} = 2 \pi f \frac{L}{R} = 2 \pi \frac{T}{t}$$

Summary
In summary RL series circuit has a first order differential equation of current
 * $$\frac{d}{dt} f(t) + \frac{1}{T} = 0$$

Which has one real root
 * $$I(t) = Ae^\frac{t}{T}$$
 * $$A = e^c$$

The Natural Response of the circuit at equilibrium is a Exponential Decrease function

Phase Angle of Difference Between Voltage and Current
 * $$Tan \theta = \omega \frac{L}{R} = 2 \pi f \frac{L}{R} = 2 \pi \frac{T}{t}$$

Natural Response
The Total Circuit's Impedance in Rectangular Form
 * $$Z = |Z| \angle \theta$$
 * $$Z = |Z_L - Z_C| \angle \pm 90$$ . ZL = ZC
 * $$Z = 0 \angle 0$$ . ZL = ZC

Circuit's Natural Response at equilibrium
 * $$L \frac{dI}{dt} + \frac{1}{C} \int I dt = 0$$
 * $$\frac{d^2I}{dt^2} + \frac{1}{LC} = 0$$
 * $$ s^2 + \frac{1}{LC} = 0$$
 * $$s = \pm \sqrt{\frac{1}{LC}} t = \pm \omega t$$
 * $$I = e^ (st)$$
 * $$I = e^j\omega t + e^ -j \omega t$$
 * $$I = A Sin \omega t$$

The Natural Response at equilibrium of the circuit is a Sinusoidal Wave

Resonance Response
At Resonance, The total Circuit's impedance is zero and the total volages are zero
 * $$Z_L - Z_C = 0$$
 * $$\omega L = \frac{1}{\omega C}$$
 * $$\omega = \sqrt{\frac{1}{LC}}$$
 * $$V_L + V_C = 0$$
 * $$V_L = - V_C $$

The Resonance Reponse of the circuit at resonance is a Standing (Sinusoidal) Wave

Natural Response

 * [[Image:RLC series circuit.png|100px]]

At Equilibrium, the sum of all voltages equal to zero
 * $$L \frac{dI}{dt} + IR + \frac{1}{C} \int I dt = 0$$
 * $$\frac{dI}{dt} + I \frac{R}{L} + \frac{1}{LC} = 0$$
 * $$\frac{d^2I}{dt^2} + \frac{R}{L} \frac{dI}{dt} + \frac{1}{LC} = 0$$
 * $$s^2 + \frac{R}{L} s + \frac{1}{LC} = 0$$
 * $$s = (-\alpha \pm \lambda) t$$

Với
 * $$\alpha = \frac{R}{2L}$$ và
 * $$\beta = \frac{1}{LC}$$
 * $$\lambda = \sqrt{\alpha^2 - \beta^2}$$

Khi $$\alpha^2 = \beta^2$$
 * $$s = -\alpha t$$
 * $$ I = e^(-\alpha)t $$


 * The response of the circuit is an Exponential Deacy

Khi $$\alpha^2 > \beta^2$$
 * $$s = (-\alpha \pm \lambda) t$$
 * $$ I = e^-\alpha t \pm (e^\lambda t + e^-\lambda t)$$


 * The response of the circuit is an Exponential Deacy

Khi $$\alpha^2 < \beta^2$$
 * $$s = (-\alpha \pm \lambda) t$$
 * $$ I = e^-\alpha t \pm (e^j\lambda t + e^-j\lambda t)$$
 * The response of the circuit is an Exponential decay sinusoidal wave

Điện Kháng Tổng Mạch Điện
 * $$Z = Z_R + Z_L + Z_C $$
 * $$Z = R + j\omega L + \frac{1}{j \omega C}$$
 * $$Z = \frac{1}{j \omega C} (j\omega^2 + j \omega \frac{R}{L} + \frac{1}{LC}) $$

Resonance Response
The total impedance of the circuit
 * $$Z = Z_R + Z_L + Z_C = R + 0 = R$$
 * $$I = \frac{V}{R}$$


 * $$Z_L = Z_C $$
 * $$j\omega L = \frac{1}{j\omega C} $$
 * $$\omega = \sqrt{\frac{1}{LC}} $$

At resonance frequency $$\omega = \sqrt{\frac{1}{LC}} $$ the total impedance of the circuit is Z = R ; at its minimum value and current will be at its maximum value  : $$I = \frac{V}{R}$$

Look at the circuit, at $$At \omega = 0 Z_C = oo$$, Capacitor opens circuit. Therefore, current is equal to zero. At $$ \omega = oo Z_L = oo$$, Inductor opens circuit. Therefore, current is equal to zero

Series RL, RC
Series RC and RL has a Character first order differential equation of the form
 * $$\frac{d f(t)}{dt} + \omega t= 0$$

that has Decay exponential function as Natural Response
 * $$f(x) = A e^(-\frac{t}{T})$$


 * f(t) = i(t) for series RL
 * f(t) = v(t) for series RC

Series LC, RLC
Series LC and RLC has a Characteristic Second order differential equation of the form
 * $$\frac{d^2 f(t)}{dt} + \omega t= 0$$
 * $$f(x) = e^(\pm \omega t)$$
 * $$f(x) = e^(\omega t) + e^(-\omega t) = A Sin \omega t$$

At equilibrium, the Natural Response of the circuit is Sinusoidal Wave
 * $$f(x) = A Sin \omega t$$

At Equilibrum, the Resonance Response is Standing Wave Reponse